Two figures are similar if they have the same shape but possibly different sizes, meaning all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).
A photo and its enlargement are similar—same shape, different size.
Showing a random 20 of 50 problems.
Example 1
medium
In triangle ABC, a line parallel to BC cuts AB at D and AC at E. If AD=4, DB=6, and AE=6, find EC.
Example 2
medium
In two similar triangles, one has sides 5,12,13 and the other has its shortest side 20. Find the longer leg of the second triangle.Triangle 1 (5-12-13 right triangle); the second similar triangle has its shortest side
Example 3
medium
Triangle A has sides 3,4,5. Triangle B has sides 9,12,16. Are they similar?Triangle A (sides 3-4-5) — check whether Triangle B (sides 9-12-16) is similar
Example 4
easy
Two similar triangles have scale factor 4. A side on the small triangle is 7. Find the matching side on the large triangle.
Example 5
medium
Triangles ABC∼XYZ with ∠A=∠X, ∠B=∠Y. If AB=10,XY=15, XZ=12, find AC.
Example 6
easy
Two similar triangles have scale factor 5. How do their areas compare?
Example 7
challenge
Explain why all circles are similar to each other, and why this means π is the same for every circle.
Example 8
hard
A tree casts a 15 m shadow at the same time a 2 m pole casts a 3 m shadow. How tall is the tree?
Example 9
medium
Triangle ABC is similar to triangle DEF. If AB=6, BC=8, AC=10, and DE=9, find EF and DF.
Example 10
easy
Two similar figures have corresponding sides 5 and 15. What is the scale factor (large to small)?
Example 11
easy
Are all equilateral triangles similar?
Example 12
medium
Two similar pyramids have a height ratio 2:7. Find the ratio of their volumes.
Example 13
hard
Two similar cones have volumes 125 and 343 cm3. Find the ratio of their slant heights.
Example 14
medium
Two similar solids have scale factor 3. The smaller has volume 40 cm3. Find the larger's volume.
Example 15
medium
Two similar triangles have corresponding side lengths in the ratio 3:5. If the perimeter of the smaller triangle is 27 cm, find the perimeter of the larger triangle.
Example 16
easy
Two similar rectangles have sides 3×5 and 9×k. Find k.
Example 17
challenge
Two similar tanks hold water. The smaller has surface area 50 m2 and capacity 40 L. The larger has linear dimensions twice the smaller. Find the larger's surface area and capacity.
Example 18
challenge
Two similar triangles have areas in the ratio 4:9. The perimeter of the smaller is 24. Find the perimeter of the larger.
Example 19
easy
True or false: two figures can be congruent but not similar.
Example 20
hard
△ADE∼△ABC with DE∥BC. AD=x, AB=12. If area of △ADE equals 1/9 of △ABC, find x.